Properties

Label 3465.2.a.bl.1.3
Level 3465
Weight 2
Character 3465.1
Self dual yes
Analytic conductor 27.668
Analytic rank 0
Dimension 4
CM no
Inner twists 1

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Newspace parameters

Level: \( N \) \(=\) \( 3465 = 3^{2} \cdot 5 \cdot 7 \cdot 11 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 3465.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(27.6681643004\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: 4.4.13448.1
Defining polynomial: \(x^{4} - 7 x^{2} + 2\)
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 2 \)
Twist minimal: no (minimal twist has level 1155)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.3
Root \(0.546295\) of defining polynomial
Character \(\chi\) \(=\) 3465.1

$q$-expansion

\(f(q)\) \(=\) \(q+0.546295 q^{2} -1.70156 q^{4} +1.00000 q^{5} -1.00000 q^{7} -2.02214 q^{8} +O(q^{10})\) \(q+0.546295 q^{2} -1.70156 q^{4} +1.00000 q^{5} -1.00000 q^{7} -2.02214 q^{8} +0.546295 q^{10} +1.00000 q^{11} +4.56844 q^{13} -0.546295 q^{14} +2.29844 q^{16} -2.70156 q^{17} -8.02362 q^{19} -1.70156 q^{20} +0.546295 q^{22} +3.52790 q^{23} +1.00000 q^{25} +2.49571 q^{26} +1.70156 q^{28} -0.133124 q^{29} +2.33897 q^{31} +5.29991 q^{32} -1.47585 q^{34} -1.00000 q^{35} +6.75362 q^{37} -4.38326 q^{38} -2.02214 q^{40} -10.9831 q^{41} +9.45518 q^{43} -1.70156 q^{44} +1.92728 q^{46} +0.649507 q^{47} +1.00000 q^{49} +0.546295 q^{50} -7.77348 q^{52} -11.8384 q^{53} +1.00000 q^{55} +2.02214 q^{56} -0.0727248 q^{58} +0.959466 q^{59} +13.1162 q^{61} +1.27777 q^{62} -1.70156 q^{64} +4.56844 q^{65} +6.41464 q^{67} +4.59688 q^{68} -0.546295 q^{70} +2.56844 q^{71} +12.5400 q^{73} +3.68947 q^{74} +13.6527 q^{76} -1.00000 q^{77} +2.56844 q^{79} +2.29844 q^{80} -6.00000 q^{82} +7.75737 q^{83} -2.70156 q^{85} +5.16531 q^{86} -2.02214 q^{88} -13.3741 q^{89} -4.56844 q^{91} -6.00295 q^{92} +0.354822 q^{94} -8.02362 q^{95} -1.78581 q^{97} +0.546295 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4q + 6q^{4} + 4q^{5} - 4q^{7} + O(q^{10}) \) \( 4q + 6q^{4} + 4q^{5} - 4q^{7} + 4q^{11} + 8q^{13} + 22q^{16} + 2q^{17} + 10q^{19} + 6q^{20} + 2q^{23} + 4q^{25} - 20q^{26} - 6q^{28} + 2q^{29} + 24q^{31} - 4q^{35} + 8q^{37} - 16q^{38} + 6q^{43} + 6q^{44} - 12q^{46} - 4q^{47} + 4q^{49} + 12q^{52} - 14q^{53} + 4q^{55} - 20q^{58} + 2q^{59} + 6q^{61} - 8q^{62} + 6q^{64} + 8q^{65} - 8q^{67} + 44q^{68} + 4q^{73} + 36q^{74} + 56q^{76} - 4q^{77} + 22q^{80} - 24q^{82} - 6q^{83} + 2q^{85} + 36q^{86} - 18q^{89} - 8q^{91} + 44q^{92} - 36q^{94} + 10q^{95} - 6q^{97} + O(q^{100}) \)

Coefficient data

For each \(n\) we display the coefficients of the \(q\)-expansion \(a_n\), the Satake parameters \(\alpha_p\), and the Satake angles \(\theta_p = \textrm{Arg}(\alpha_p)\).

Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0.546295 0.386289 0.193144 0.981170i \(-0.438131\pi\)
0.193144 + 0.981170i \(0.438131\pi\)
\(3\) 0 0
\(4\) −1.70156 −0.850781
\(5\) 1.00000 0.447214
\(6\) 0 0
\(7\) −1.00000 −0.377964
\(8\) −2.02214 −0.714936
\(9\) 0 0
\(10\) 0.546295 0.172754
\(11\) 1.00000 0.301511
\(12\) 0 0
\(13\) 4.56844 1.26706 0.633528 0.773719i \(-0.281606\pi\)
0.633528 + 0.773719i \(0.281606\pi\)
\(14\) −0.546295 −0.146003
\(15\) 0 0
\(16\) 2.29844 0.574609
\(17\) −2.70156 −0.655225 −0.327613 0.944812i \(-0.606244\pi\)
−0.327613 + 0.944812i \(0.606244\pi\)
\(18\) 0 0
\(19\) −8.02362 −1.84074 −0.920372 0.391044i \(-0.872114\pi\)
−0.920372 + 0.391044i \(0.872114\pi\)
\(20\) −1.70156 −0.380481
\(21\) 0 0
\(22\) 0.546295 0.116470
\(23\) 3.52790 0.735619 0.367809 0.929901i \(-0.380108\pi\)
0.367809 + 0.929901i \(0.380108\pi\)
\(24\) 0 0
\(25\) 1.00000 0.200000
\(26\) 2.49571 0.489450
\(27\) 0 0
\(28\) 1.70156 0.321565
\(29\) −0.133124 −0.0247205 −0.0123602 0.999924i \(-0.503934\pi\)
−0.0123602 + 0.999924i \(0.503934\pi\)
\(30\) 0 0
\(31\) 2.33897 0.420092 0.210046 0.977692i \(-0.432639\pi\)
0.210046 + 0.977692i \(0.432639\pi\)
\(32\) 5.29991 0.936901
\(33\) 0 0
\(34\) −1.47585 −0.253106
\(35\) −1.00000 −0.169031
\(36\) 0 0
\(37\) 6.75362 1.11029 0.555144 0.831754i \(-0.312663\pi\)
0.555144 + 0.831754i \(0.312663\pi\)
\(38\) −4.38326 −0.711059
\(39\) 0 0
\(40\) −2.02214 −0.319729
\(41\) −10.9831 −1.71527 −0.857635 0.514259i \(-0.828067\pi\)
−0.857635 + 0.514259i \(0.828067\pi\)
\(42\) 0 0
\(43\) 9.45518 1.44190 0.720951 0.692986i \(-0.243705\pi\)
0.720951 + 0.692986i \(0.243705\pi\)
\(44\) −1.70156 −0.256520
\(45\) 0 0
\(46\) 1.92728 0.284161
\(47\) 0.649507 0.0947403 0.0473702 0.998877i \(-0.484916\pi\)
0.0473702 + 0.998877i \(0.484916\pi\)
\(48\) 0 0
\(49\) 1.00000 0.142857
\(50\) 0.546295 0.0772577
\(51\) 0 0
\(52\) −7.77348 −1.07799
\(53\) −11.8384 −1.62613 −0.813067 0.582170i \(-0.802204\pi\)
−0.813067 + 0.582170i \(0.802204\pi\)
\(54\) 0 0
\(55\) 1.00000 0.134840
\(56\) 2.02214 0.270220
\(57\) 0 0
\(58\) −0.0727248 −0.00954923
\(59\) 0.959466 0.124912 0.0624559 0.998048i \(-0.480107\pi\)
0.0624559 + 0.998048i \(0.480107\pi\)
\(60\) 0 0
\(61\) 13.1162 1.67936 0.839679 0.543083i \(-0.182743\pi\)
0.839679 + 0.543083i \(0.182743\pi\)
\(62\) 1.27777 0.162277
\(63\) 0 0
\(64\) −1.70156 −0.212695
\(65\) 4.56844 0.566645
\(66\) 0 0
\(67\) 6.41464 0.783674 0.391837 0.920035i \(-0.371840\pi\)
0.391837 + 0.920035i \(0.371840\pi\)
\(68\) 4.59688 0.557453
\(69\) 0 0
\(70\) −0.546295 −0.0652947
\(71\) 2.56844 0.304818 0.152409 0.988318i \(-0.451297\pi\)
0.152409 + 0.988318i \(0.451297\pi\)
\(72\) 0 0
\(73\) 12.5400 1.46770 0.733848 0.679314i \(-0.237722\pi\)
0.733848 + 0.679314i \(0.237722\pi\)
\(74\) 3.68947 0.428892
\(75\) 0 0
\(76\) 13.6527 1.56607
\(77\) −1.00000 −0.113961
\(78\) 0 0
\(79\) 2.56844 0.288972 0.144486 0.989507i \(-0.453847\pi\)
0.144486 + 0.989507i \(0.453847\pi\)
\(80\) 2.29844 0.256973
\(81\) 0 0
\(82\) −6.00000 −0.662589
\(83\) 7.75737 0.851482 0.425741 0.904845i \(-0.360013\pi\)
0.425741 + 0.904845i \(0.360013\pi\)
\(84\) 0 0
\(85\) −2.70156 −0.293026
\(86\) 5.16531 0.556990
\(87\) 0 0
\(88\) −2.02214 −0.215561
\(89\) −13.3741 −1.41765 −0.708826 0.705383i \(-0.750775\pi\)
−0.708826 + 0.705383i \(0.750775\pi\)
\(90\) 0 0
\(91\) −4.56844 −0.478902
\(92\) −6.00295 −0.625851
\(93\) 0 0
\(94\) 0.354822 0.0365971
\(95\) −8.02362 −0.823206
\(96\) 0 0
\(97\) −1.78581 −0.181321 −0.0906606 0.995882i \(-0.528898\pi\)
−0.0906606 + 0.995882i \(0.528898\pi\)
\(98\) 0.546295 0.0551841
\(99\) 0 0
\(100\) −1.70156 −0.170156
\(101\) 11.6326 1.15749 0.578743 0.815510i \(-0.303543\pi\)
0.578743 + 0.815510i \(0.303543\pi\)
\(102\) 0 0
\(103\) −9.18893 −0.905412 −0.452706 0.891660i \(-0.649541\pi\)
−0.452706 + 0.891660i \(0.649541\pi\)
\(104\) −9.23804 −0.905864
\(105\) 0 0
\(106\) −6.46728 −0.628157
\(107\) 13.0199 1.25868 0.629339 0.777131i \(-0.283326\pi\)
0.629339 + 0.777131i \(0.283326\pi\)
\(108\) 0 0
\(109\) 0.109506 0.0104888 0.00524439 0.999986i \(-0.498331\pi\)
0.00524439 + 0.999986i \(0.498331\pi\)
\(110\) 0.546295 0.0520872
\(111\) 0 0
\(112\) −2.29844 −0.217182
\(113\) 3.83844 0.361090 0.180545 0.983567i \(-0.442214\pi\)
0.180545 + 0.983567i \(0.442214\pi\)
\(114\) 0 0
\(115\) 3.52790 0.328979
\(116\) 0.226518 0.0210317
\(117\) 0 0
\(118\) 0.524151 0.0482520
\(119\) 2.70156 0.247652
\(120\) 0 0
\(121\) 1.00000 0.0909091
\(122\) 7.16531 0.648717
\(123\) 0 0
\(124\) −3.97991 −0.357406
\(125\) 1.00000 0.0894427
\(126\) 0 0
\(127\) −5.45518 −0.484069 −0.242034 0.970268i \(-0.577815\pi\)
−0.242034 + 0.970268i \(0.577815\pi\)
\(128\) −11.5294 −1.01906
\(129\) 0 0
\(130\) 2.49571 0.218889
\(131\) 21.1084 1.84425 0.922126 0.386889i \(-0.126450\pi\)
0.922126 + 0.386889i \(0.126450\pi\)
\(132\) 0 0
\(133\) 8.02362 0.695736
\(134\) 3.50429 0.302724
\(135\) 0 0
\(136\) 5.46295 0.468444
\(137\) 3.40312 0.290749 0.145374 0.989377i \(-0.453561\pi\)
0.145374 + 0.989377i \(0.453561\pi\)
\(138\) 0 0
\(139\) 21.2410 1.80164 0.900818 0.434196i \(-0.142967\pi\)
0.900818 + 0.434196i \(0.142967\pi\)
\(140\) 1.70156 0.143808
\(141\) 0 0
\(142\) 1.40312 0.117748
\(143\) 4.56844 0.382032
\(144\) 0 0
\(145\) −0.133124 −0.0110553
\(146\) 6.85054 0.566954
\(147\) 0 0
\(148\) −11.4917 −0.944612
\(149\) 11.5072 0.942709 0.471355 0.881944i \(-0.343765\pi\)
0.471355 + 0.881944i \(0.343765\pi\)
\(150\) 0 0
\(151\) 7.58830 0.617527 0.308764 0.951139i \(-0.400085\pi\)
0.308764 + 0.951139i \(0.400085\pi\)
\(152\) 16.2249 1.31601
\(153\) 0 0
\(154\) −0.546295 −0.0440217
\(155\) 2.33897 0.187871
\(156\) 0 0
\(157\) −12.8583 −1.02620 −0.513102 0.858328i \(-0.671504\pi\)
−0.513102 + 0.858328i \(0.671504\pi\)
\(158\) 1.40312 0.111627
\(159\) 0 0
\(160\) 5.29991 0.418995
\(161\) −3.52790 −0.278038
\(162\) 0 0
\(163\) 22.6525 1.77428 0.887139 0.461503i \(-0.152690\pi\)
0.887139 + 0.461503i \(0.152690\pi\)
\(164\) 18.6884 1.45932
\(165\) 0 0
\(166\) 4.23781 0.328918
\(167\) 3.21795 0.249012 0.124506 0.992219i \(-0.460265\pi\)
0.124506 + 0.992219i \(0.460265\pi\)
\(168\) 0 0
\(169\) 7.87063 0.605433
\(170\) −1.47585 −0.113192
\(171\) 0 0
\(172\) −16.0886 −1.22674
\(173\) 18.2094 1.38443 0.692216 0.721690i \(-0.256634\pi\)
0.692216 + 0.721690i \(0.256634\pi\)
\(174\) 0 0
\(175\) −1.00000 −0.0755929
\(176\) 2.29844 0.173251
\(177\) 0 0
\(178\) −7.30621 −0.547623
\(179\) 4.75362 0.355302 0.177651 0.984094i \(-0.443150\pi\)
0.177651 + 0.984094i \(0.443150\pi\)
\(180\) 0 0
\(181\) 0.347316 0.0258158 0.0129079 0.999917i \(-0.495891\pi\)
0.0129079 + 0.999917i \(0.495891\pi\)
\(182\) −2.49571 −0.184995
\(183\) 0 0
\(184\) −7.13393 −0.525920
\(185\) 6.75362 0.496536
\(186\) 0 0
\(187\) −2.70156 −0.197558
\(188\) −1.10518 −0.0806033
\(189\) 0 0
\(190\) −4.38326 −0.317995
\(191\) −11.8379 −0.856558 −0.428279 0.903647i \(-0.640880\pi\)
−0.428279 + 0.903647i \(0.640880\pi\)
\(192\) 0 0
\(193\) −4.14840 −0.298608 −0.149304 0.988791i \(-0.547703\pi\)
−0.149304 + 0.988791i \(0.547703\pi\)
\(194\) −0.975577 −0.0700424
\(195\) 0 0
\(196\) −1.70156 −0.121540
\(197\) −19.3663 −1.37979 −0.689897 0.723907i \(-0.742344\pi\)
−0.689897 + 0.723907i \(0.742344\pi\)
\(198\) 0 0
\(199\) 13.2622 0.940135 0.470067 0.882631i \(-0.344230\pi\)
0.470067 + 0.882631i \(0.344230\pi\)
\(200\) −2.02214 −0.142987
\(201\) 0 0
\(202\) 6.35482 0.447124
\(203\) 0.133124 0.00934345
\(204\) 0 0
\(205\) −10.9831 −0.767092
\(206\) −5.01986 −0.349751
\(207\) 0 0
\(208\) 10.5003 0.728063
\(209\) −8.02362 −0.555005
\(210\) 0 0
\(211\) 2.54000 0.174861 0.0874304 0.996171i \(-0.472134\pi\)
0.0874304 + 0.996171i \(0.472134\pi\)
\(212\) 20.1438 1.38348
\(213\) 0 0
\(214\) 7.11268 0.486213
\(215\) 9.45518 0.644838
\(216\) 0 0
\(217\) −2.33897 −0.158780
\(218\) 0.0598226 0.00405170
\(219\) 0 0
\(220\) −1.70156 −0.114719
\(221\) −12.3419 −0.830207
\(222\) 0 0
\(223\) 20.5110 1.37352 0.686759 0.726886i \(-0.259033\pi\)
0.686759 + 0.726886i \(0.259033\pi\)
\(224\) −5.29991 −0.354115
\(225\) 0 0
\(226\) 2.09692 0.139485
\(227\) −19.6119 −1.30169 −0.650844 0.759211i \(-0.725585\pi\)
−0.650844 + 0.759211i \(0.725585\pi\)
\(228\) 0 0
\(229\) 20.4230 1.34959 0.674795 0.738006i \(-0.264232\pi\)
0.674795 + 0.738006i \(0.264232\pi\)
\(230\) 1.92728 0.127081
\(231\) 0 0
\(232\) 0.269195 0.0176735
\(233\) 17.3304 1.13535 0.567676 0.823252i \(-0.307843\pi\)
0.567676 + 0.823252i \(0.307843\pi\)
\(234\) 0 0
\(235\) 0.649507 0.0423692
\(236\) −1.63259 −0.106273
\(237\) 0 0
\(238\) 1.47585 0.0956651
\(239\) 11.6404 0.752952 0.376476 0.926426i \(-0.377136\pi\)
0.376476 + 0.926426i \(0.377136\pi\)
\(240\) 0 0
\(241\) −22.5429 −1.45212 −0.726059 0.687632i \(-0.758650\pi\)
−0.726059 + 0.687632i \(0.758650\pi\)
\(242\) 0.546295 0.0351172
\(243\) 0 0
\(244\) −22.3180 −1.42877
\(245\) 1.00000 0.0638877
\(246\) 0 0
\(247\) −36.6554 −2.33233
\(248\) −4.72974 −0.300339
\(249\) 0 0
\(250\) 0.546295 0.0345507
\(251\) −11.4474 −0.722554 −0.361277 0.932458i \(-0.617659\pi\)
−0.361277 + 0.932458i \(0.617659\pi\)
\(252\) 0 0
\(253\) 3.52790 0.221797
\(254\) −2.98014 −0.186990
\(255\) 0 0
\(256\) −2.89531 −0.180957
\(257\) 28.6157 1.78500 0.892498 0.451051i \(-0.148951\pi\)
0.892498 + 0.451051i \(0.148951\pi\)
\(258\) 0 0
\(259\) −6.75362 −0.419649
\(260\) −7.77348 −0.482091
\(261\) 0 0
\(262\) 11.5314 0.712414
\(263\) 8.67794 0.535105 0.267552 0.963543i \(-0.413785\pi\)
0.267552 + 0.963543i \(0.413785\pi\)
\(264\) 0 0
\(265\) −11.8384 −0.727230
\(266\) 4.38326 0.268755
\(267\) 0 0
\(268\) −10.9149 −0.666735
\(269\) −29.8911 −1.82249 −0.911245 0.411864i \(-0.864878\pi\)
−0.911245 + 0.411864i \(0.864878\pi\)
\(270\) 0 0
\(271\) 20.9678 1.27370 0.636852 0.770986i \(-0.280236\pi\)
0.636852 + 0.770986i \(0.280236\pi\)
\(272\) −6.20937 −0.376499
\(273\) 0 0
\(274\) 1.85911 0.112313
\(275\) 1.00000 0.0603023
\(276\) 0 0
\(277\) −13.7137 −0.823974 −0.411987 0.911190i \(-0.635165\pi\)
−0.411987 + 0.911190i \(0.635165\pi\)
\(278\) 11.6038 0.695952
\(279\) 0 0
\(280\) 2.02214 0.120846
\(281\) −4.91036 −0.292927 −0.146464 0.989216i \(-0.546789\pi\)
−0.146464 + 0.989216i \(0.546789\pi\)
\(282\) 0 0
\(283\) −25.4788 −1.51456 −0.757279 0.653092i \(-0.773472\pi\)
−0.757279 + 0.653092i \(0.773472\pi\)
\(284\) −4.37036 −0.259333
\(285\) 0 0
\(286\) 2.49571 0.147575
\(287\) 10.9831 0.648311
\(288\) 0 0
\(289\) −9.70156 −0.570680
\(290\) −0.0727248 −0.00427055
\(291\) 0 0
\(292\) −21.3376 −1.24869
\(293\) 2.16907 0.126718 0.0633591 0.997991i \(-0.479819\pi\)
0.0633591 + 0.997991i \(0.479819\pi\)
\(294\) 0 0
\(295\) 0.959466 0.0558622
\(296\) −13.6568 −0.793784
\(297\) 0 0
\(298\) 6.28634 0.364158
\(299\) 16.1170 0.932071
\(300\) 0 0
\(301\) −9.45518 −0.544987
\(302\) 4.14545 0.238544
\(303\) 0 0
\(304\) −18.4418 −1.05771
\(305\) 13.1162 0.751032
\(306\) 0 0
\(307\) 12.3548 0.705127 0.352563 0.935788i \(-0.385310\pi\)
0.352563 + 0.935788i \(0.385310\pi\)
\(308\) 1.70156 0.0969555
\(309\) 0 0
\(310\) 1.27777 0.0725724
\(311\) −22.6073 −1.28194 −0.640972 0.767564i \(-0.721469\pi\)
−0.640972 + 0.767564i \(0.721469\pi\)
\(312\) 0 0
\(313\) −34.2614 −1.93657 −0.968285 0.249848i \(-0.919620\pi\)
−0.968285 + 0.249848i \(0.919620\pi\)
\(314\) −7.02442 −0.396411
\(315\) 0 0
\(316\) −4.37036 −0.245852
\(317\) −8.28929 −0.465573 −0.232786 0.972528i \(-0.574784\pi\)
−0.232786 + 0.972528i \(0.574784\pi\)
\(318\) 0 0
\(319\) −0.133124 −0.00745350
\(320\) −1.70156 −0.0951202
\(321\) 0 0
\(322\) −1.92728 −0.107403
\(323\) 21.6763 1.20610
\(324\) 0 0
\(325\) 4.56844 0.253411
\(326\) 12.3749 0.685383
\(327\) 0 0
\(328\) 22.2094 1.22631
\(329\) −0.649507 −0.0358085
\(330\) 0 0
\(331\) 21.4267 1.17772 0.588860 0.808235i \(-0.299577\pi\)
0.588860 + 0.808235i \(0.299577\pi\)
\(332\) −13.1996 −0.724425
\(333\) 0 0
\(334\) 1.75795 0.0961906
\(335\) 6.41464 0.350469
\(336\) 0 0
\(337\) −20.2585 −1.10355 −0.551775 0.833993i \(-0.686049\pi\)
−0.551775 + 0.833993i \(0.686049\pi\)
\(338\) 4.29968 0.233872
\(339\) 0 0
\(340\) 4.59688 0.249301
\(341\) 2.33897 0.126662
\(342\) 0 0
\(343\) −1.00000 −0.0539949
\(344\) −19.1197 −1.03087
\(345\) 0 0
\(346\) 9.94768 0.534791
\(347\) 23.9431 1.28533 0.642667 0.766146i \(-0.277828\pi\)
0.642667 + 0.766146i \(0.277828\pi\)
\(348\) 0 0
\(349\) −25.5751 −1.36901 −0.684503 0.729010i \(-0.739981\pi\)
−0.684503 + 0.729010i \(0.739981\pi\)
\(350\) −0.546295 −0.0292007
\(351\) 0 0
\(352\) 5.29991 0.282486
\(353\) −0.330628 −0.0175976 −0.00879878 0.999961i \(-0.502801\pi\)
−0.00879878 + 0.999961i \(0.502801\pi\)
\(354\) 0 0
\(355\) 2.56844 0.136319
\(356\) 22.7569 1.20611
\(357\) 0 0
\(358\) 2.59688 0.137249
\(359\) 23.6248 1.24687 0.623435 0.781875i \(-0.285737\pi\)
0.623435 + 0.781875i \(0.285737\pi\)
\(360\) 0 0
\(361\) 45.3784 2.38834
\(362\) 0.189737 0.00997235
\(363\) 0 0
\(364\) 7.77348 0.407441
\(365\) 12.5400 0.656374
\(366\) 0 0
\(367\) −22.3730 −1.16786 −0.583932 0.811803i \(-0.698486\pi\)
−0.583932 + 0.811803i \(0.698486\pi\)
\(368\) 8.10867 0.422694
\(369\) 0 0
\(370\) 3.68947 0.191806
\(371\) 11.8384 0.614621
\(372\) 0 0
\(373\) −11.9922 −0.620934 −0.310467 0.950584i \(-0.600485\pi\)
−0.310467 + 0.950584i \(0.600485\pi\)
\(374\) −1.47585 −0.0763143
\(375\) 0 0
\(376\) −1.31340 −0.0677333
\(377\) −0.608168 −0.0313222
\(378\) 0 0
\(379\) 18.2088 0.935323 0.467662 0.883908i \(-0.345097\pi\)
0.467662 + 0.883908i \(0.345097\pi\)
\(380\) 13.6527 0.700368
\(381\) 0 0
\(382\) −6.46696 −0.330879
\(383\) −22.3494 −1.14200 −0.571001 0.820949i \(-0.693445\pi\)
−0.571001 + 0.820949i \(0.693445\pi\)
\(384\) 0 0
\(385\) −1.00000 −0.0509647
\(386\) −2.26625 −0.115349
\(387\) 0 0
\(388\) 3.03866 0.154265
\(389\) −25.4863 −1.29221 −0.646103 0.763250i \(-0.723603\pi\)
−0.646103 + 0.763250i \(0.723603\pi\)
\(390\) 0 0
\(391\) −9.53085 −0.481996
\(392\) −2.02214 −0.102134
\(393\) 0 0
\(394\) −10.5797 −0.532999
\(395\) 2.56844 0.129232
\(396\) 0 0
\(397\) 10.3548 0.519694 0.259847 0.965650i \(-0.416328\pi\)
0.259847 + 0.965650i \(0.416328\pi\)
\(398\) 7.24509 0.363163
\(399\) 0 0
\(400\) 2.29844 0.114922
\(401\) −0.0939709 −0.00469268 −0.00234634 0.999997i \(-0.500747\pi\)
−0.00234634 + 0.999997i \(0.500747\pi\)
\(402\) 0 0
\(403\) 10.6855 0.532280
\(404\) −19.7936 −0.984767
\(405\) 0 0
\(406\) 0.0727248 0.00360927
\(407\) 6.75362 0.334764
\(408\) 0 0
\(409\) 2.90741 0.143762 0.0718811 0.997413i \(-0.477100\pi\)
0.0718811 + 0.997413i \(0.477100\pi\)
\(410\) −6.00000 −0.296319
\(411\) 0 0
\(412\) 15.6355 0.770308
\(413\) −0.959466 −0.0472122
\(414\) 0 0
\(415\) 7.75737 0.380794
\(416\) 24.2123 1.18711
\(417\) 0 0
\(418\) −4.38326 −0.214392
\(419\) −16.2660 −0.794645 −0.397323 0.917679i \(-0.630061\pi\)
−0.397323 + 0.917679i \(0.630061\pi\)
\(420\) 0 0
\(421\) −16.1047 −0.784894 −0.392447 0.919775i \(-0.628371\pi\)
−0.392447 + 0.919775i \(0.628371\pi\)
\(422\) 1.38759 0.0675468
\(423\) 0 0
\(424\) 23.9390 1.16258
\(425\) −2.70156 −0.131045
\(426\) 0 0
\(427\) −13.1162 −0.634738
\(428\) −22.1541 −1.07086
\(429\) 0 0
\(430\) 5.16531 0.249094
\(431\) 19.3221 0.930711 0.465355 0.885124i \(-0.345927\pi\)
0.465355 + 0.885124i \(0.345927\pi\)
\(432\) 0 0
\(433\) 37.0462 1.78033 0.890163 0.455643i \(-0.150591\pi\)
0.890163 + 0.455643i \(0.150591\pi\)
\(434\) −1.27777 −0.0613348
\(435\) 0 0
\(436\) −0.186331 −0.00892366
\(437\) −28.3066 −1.35409
\(438\) 0 0
\(439\) 14.4750 0.690856 0.345428 0.938445i \(-0.387734\pi\)
0.345428 + 0.938445i \(0.387734\pi\)
\(440\) −2.02214 −0.0964019
\(441\) 0 0
\(442\) −6.74233 −0.320700
\(443\) 26.4146 1.25500 0.627499 0.778618i \(-0.284079\pi\)
0.627499 + 0.778618i \(0.284079\pi\)
\(444\) 0 0
\(445\) −13.3741 −0.633994
\(446\) 11.2050 0.530574
\(447\) 0 0
\(448\) 1.70156 0.0803913
\(449\) 13.7526 0.649023 0.324511 0.945882i \(-0.394800\pi\)
0.324511 + 0.945882i \(0.394800\pi\)
\(450\) 0 0
\(451\) −10.9831 −0.517173
\(452\) −6.53134 −0.307208
\(453\) 0 0
\(454\) −10.7139 −0.502828
\(455\) −4.56844 −0.214172
\(456\) 0 0
\(457\) −9.98473 −0.467066 −0.233533 0.972349i \(-0.575029\pi\)
−0.233533 + 0.972349i \(0.575029\pi\)
\(458\) 11.1570 0.521331
\(459\) 0 0
\(460\) −6.00295 −0.279889
\(461\) −3.31911 −0.154586 −0.0772931 0.997008i \(-0.524628\pi\)
−0.0772931 + 0.997008i \(0.524628\pi\)
\(462\) 0 0
\(463\) −20.1726 −0.937500 −0.468750 0.883331i \(-0.655295\pi\)
−0.468750 + 0.883331i \(0.655295\pi\)
\(464\) −0.305977 −0.0142046
\(465\) 0 0
\(466\) 9.46751 0.438574
\(467\) −3.69241 −0.170864 −0.0854322 0.996344i \(-0.527227\pi\)
−0.0854322 + 0.996344i \(0.527227\pi\)
\(468\) 0 0
\(469\) −6.41464 −0.296201
\(470\) 0.354822 0.0163667
\(471\) 0 0
\(472\) −1.94018 −0.0893039
\(473\) 9.45518 0.434750
\(474\) 0 0
\(475\) −8.02362 −0.368149
\(476\) −4.59688 −0.210697
\(477\) 0 0
\(478\) 6.35907 0.290857
\(479\) −41.0516 −1.87569 −0.937847 0.347049i \(-0.887184\pi\)
−0.937847 + 0.347049i \(0.887184\pi\)
\(480\) 0 0
\(481\) 30.8535 1.40680
\(482\) −12.3151 −0.560937
\(483\) 0 0
\(484\) −1.70156 −0.0773437
\(485\) −1.78581 −0.0810893
\(486\) 0 0
\(487\) −8.08402 −0.366322 −0.183161 0.983083i \(-0.558633\pi\)
−0.183161 + 0.983083i \(0.558633\pi\)
\(488\) −26.5229 −1.20063
\(489\) 0 0
\(490\) 0.546295 0.0246791
\(491\) 3.47348 0.156756 0.0783779 0.996924i \(-0.475026\pi\)
0.0783779 + 0.996924i \(0.475026\pi\)
\(492\) 0 0
\(493\) 0.359642 0.0161975
\(494\) −20.0247 −0.900952
\(495\) 0 0
\(496\) 5.37598 0.241389
\(497\) −2.56844 −0.115210
\(498\) 0 0
\(499\) −29.0719 −1.30144 −0.650719 0.759319i \(-0.725532\pi\)
−0.650719 + 0.759319i \(0.725532\pi\)
\(500\) −1.70156 −0.0760962
\(501\) 0 0
\(502\) −6.25366 −0.279115
\(503\) −30.6109 −1.36487 −0.682435 0.730946i \(-0.739079\pi\)
−0.682435 + 0.730946i \(0.739079\pi\)
\(504\) 0 0
\(505\) 11.6326 0.517643
\(506\) 1.92728 0.0856778
\(507\) 0 0
\(508\) 9.28233 0.411837
\(509\) 0.696166 0.0308570 0.0154285 0.999881i \(-0.495089\pi\)
0.0154285 + 0.999881i \(0.495089\pi\)
\(510\) 0 0
\(511\) −12.5400 −0.554737
\(512\) 21.4771 0.949161
\(513\) 0 0
\(514\) 15.6326 0.689524
\(515\) −9.18893 −0.404913
\(516\) 0 0
\(517\) 0.649507 0.0285653
\(518\) −3.68947 −0.162106
\(519\) 0 0
\(520\) −9.23804 −0.405115
\(521\) −14.2142 −0.622735 −0.311368 0.950290i \(-0.600787\pi\)
−0.311368 + 0.950290i \(0.600787\pi\)
\(522\) 0 0
\(523\) 5.04830 0.220747 0.110373 0.993890i \(-0.464795\pi\)
0.110373 + 0.993890i \(0.464795\pi\)
\(524\) −35.9173 −1.56906
\(525\) 0 0
\(526\) 4.74071 0.206705
\(527\) −6.31888 −0.275255
\(528\) 0 0
\(529\) −10.5539 −0.458865
\(530\) −6.46728 −0.280921
\(531\) 0 0
\(532\) −13.6527 −0.591919
\(533\) −50.1755 −2.17334
\(534\) 0 0
\(535\) 13.0199 0.562898
\(536\) −12.9713 −0.560276
\(537\) 0 0
\(538\) −16.3293 −0.704008
\(539\) 1.00000 0.0430730
\(540\) 0 0
\(541\) 36.1154 1.55272 0.776361 0.630288i \(-0.217063\pi\)
0.776361 + 0.630288i \(0.217063\pi\)
\(542\) 11.4546 0.492017
\(543\) 0 0
\(544\) −14.3180 −0.613881
\(545\) 0.109506 0.00469073
\(546\) 0 0
\(547\) 11.4213 0.488341 0.244171 0.969732i \(-0.421484\pi\)
0.244171 + 0.969732i \(0.421484\pi\)
\(548\) −5.79063 −0.247363
\(549\) 0 0
\(550\) 0.546295 0.0232941
\(551\) 1.06813 0.0455040
\(552\) 0 0
\(553\) −2.56844 −0.109221
\(554\) −7.49170 −0.318292
\(555\) 0 0
\(556\) −36.1429 −1.53280
\(557\) −3.52733 −0.149458 −0.0747288 0.997204i \(-0.523809\pi\)
−0.0747288 + 0.997204i \(0.523809\pi\)
\(558\) 0 0
\(559\) 43.1954 1.82697
\(560\) −2.29844 −0.0971267
\(561\) 0 0
\(562\) −2.68250 −0.113155
\(563\) 10.5400 0.444208 0.222104 0.975023i \(-0.428708\pi\)
0.222104 + 0.975023i \(0.428708\pi\)
\(564\) 0 0
\(565\) 3.83844 0.161484
\(566\) −13.9189 −0.585056
\(567\) 0 0
\(568\) −5.19375 −0.217925
\(569\) 15.6173 0.654712 0.327356 0.944901i \(-0.393842\pi\)
0.327356 + 0.944901i \(0.393842\pi\)
\(570\) 0 0
\(571\) −9.94737 −0.416284 −0.208142 0.978099i \(-0.566742\pi\)
−0.208142 + 0.978099i \(0.566742\pi\)
\(572\) −7.77348 −0.325026
\(573\) 0 0
\(574\) 6.00000 0.250435
\(575\) 3.52790 0.147124
\(576\) 0 0
\(577\) 29.1030 1.21158 0.605788 0.795626i \(-0.292858\pi\)
0.605788 + 0.795626i \(0.292858\pi\)
\(578\) −5.29991 −0.220447
\(579\) 0 0
\(580\) 0.226518 0.00940566
\(581\) −7.75737 −0.321830
\(582\) 0 0
\(583\) −11.8384 −0.490298
\(584\) −25.3577 −1.04931
\(585\) 0 0
\(586\) 1.18495 0.0489498
\(587\) −12.7536 −0.526398 −0.263199 0.964742i \(-0.584778\pi\)
−0.263199 + 0.964742i \(0.584778\pi\)
\(588\) 0 0
\(589\) −18.7670 −0.773282
\(590\) 0.524151 0.0215790
\(591\) 0 0
\(592\) 15.5228 0.637982
\(593\) 3.91893 0.160931 0.0804656 0.996757i \(-0.474359\pi\)
0.0804656 + 0.996757i \(0.474359\pi\)
\(594\) 0 0
\(595\) 2.70156 0.110753
\(596\) −19.5803 −0.802039
\(597\) 0 0
\(598\) 8.80464 0.360048
\(599\) 11.9420 0.487936 0.243968 0.969783i \(-0.421551\pi\)
0.243968 + 0.969783i \(0.421551\pi\)
\(600\) 0 0
\(601\) 16.6878 0.680710 0.340355 0.940297i \(-0.389453\pi\)
0.340355 + 0.940297i \(0.389453\pi\)
\(602\) −5.16531 −0.210522
\(603\) 0 0
\(604\) −12.9120 −0.525381
\(605\) 1.00000 0.0406558
\(606\) 0 0
\(607\) 4.72518 0.191789 0.0958946 0.995391i \(-0.469429\pi\)
0.0958946 + 0.995391i \(0.469429\pi\)
\(608\) −42.5245 −1.72459
\(609\) 0 0
\(610\) 7.16531 0.290115
\(611\) 2.96723 0.120041
\(612\) 0 0
\(613\) −6.84938 −0.276644 −0.138322 0.990387i \(-0.544171\pi\)
−0.138322 + 0.990387i \(0.544171\pi\)
\(614\) 6.74937 0.272383
\(615\) 0 0
\(616\) 2.02214 0.0814745
\(617\) −21.0725 −0.848347 −0.424173 0.905581i \(-0.639435\pi\)
−0.424173 + 0.905581i \(0.639435\pi\)
\(618\) 0 0
\(619\) −29.2209 −1.17449 −0.587243 0.809410i \(-0.699787\pi\)
−0.587243 + 0.809410i \(0.699787\pi\)
\(620\) −3.97991 −0.159837
\(621\) 0 0
\(622\) −12.3503 −0.495200
\(623\) 13.3741 0.535822
\(624\) 0 0
\(625\) 1.00000 0.0400000
\(626\) −18.7168 −0.748075
\(627\) 0 0
\(628\) 21.8792 0.873075
\(629\) −18.2453 −0.727488
\(630\) 0 0
\(631\) −23.8201 −0.948265 −0.474132 0.880454i \(-0.657238\pi\)
−0.474132 + 0.880454i \(0.657238\pi\)
\(632\) −5.19375 −0.206596
\(633\) 0 0
\(634\) −4.52839 −0.179846
\(635\) −5.45518 −0.216482
\(636\) 0 0
\(637\) 4.56844 0.181008
\(638\) −0.0727248 −0.00287920
\(639\) 0 0
\(640\) −11.5294 −0.455739
\(641\) 30.9179 1.22118 0.610591 0.791946i \(-0.290932\pi\)
0.610591 + 0.791946i \(0.290932\pi\)
\(642\) 0 0
\(643\) 12.4224 0.489892 0.244946 0.969537i \(-0.421230\pi\)
0.244946 + 0.969537i \(0.421230\pi\)
\(644\) 6.00295 0.236549
\(645\) 0 0
\(646\) 11.8416 0.465903
\(647\) −31.6924 −1.24596 −0.622979 0.782239i \(-0.714078\pi\)
−0.622979 + 0.782239i \(0.714078\pi\)
\(648\) 0 0
\(649\) 0.959466 0.0376623
\(650\) 2.49571 0.0978899
\(651\) 0 0
\(652\) −38.5446 −1.50952
\(653\) −25.4037 −0.994124 −0.497062 0.867715i \(-0.665588\pi\)
−0.497062 + 0.867715i \(0.665588\pi\)
\(654\) 0 0
\(655\) 21.1084 0.824775
\(656\) −25.2439 −0.985610
\(657\) 0 0
\(658\) −0.354822 −0.0138324
\(659\) 17.1004 0.666135 0.333068 0.942903i \(-0.391916\pi\)
0.333068 + 0.942903i \(0.391916\pi\)
\(660\) 0 0
\(661\) 17.8675 0.694963 0.347482 0.937687i \(-0.387037\pi\)
0.347482 + 0.937687i \(0.387037\pi\)
\(662\) 11.7053 0.454940
\(663\) 0 0
\(664\) −15.6865 −0.608755
\(665\) 8.02362 0.311143
\(666\) 0 0
\(667\) −0.469648 −0.0181848
\(668\) −5.47553 −0.211855
\(669\) 0 0
\(670\) 3.50429 0.135382
\(671\) 13.1162 0.506346
\(672\) 0 0
\(673\) −30.7973 −1.18715 −0.593575 0.804779i \(-0.702284\pi\)
−0.593575 + 0.804779i \(0.702284\pi\)
\(674\) −11.0671 −0.426289
\(675\) 0 0
\(676\) −13.3924 −0.515091
\(677\) −1.54800 −0.0594944 −0.0297472 0.999557i \(-0.509470\pi\)
−0.0297472 + 0.999557i \(0.509470\pi\)
\(678\) 0 0
\(679\) 1.78581 0.0685330
\(680\) 5.46295 0.209494
\(681\) 0 0
\(682\) 1.27777 0.0489283
\(683\) 28.2767 1.08198 0.540989 0.841030i \(-0.318050\pi\)
0.540989 + 0.841030i \(0.318050\pi\)
\(684\) 0 0
\(685\) 3.40312 0.130027
\(686\) −0.546295 −0.0208576
\(687\) 0 0
\(688\) 21.7321 0.828530
\(689\) −54.0832 −2.06041
\(690\) 0 0
\(691\) −11.4061 −0.433907 −0.216954 0.976182i \(-0.569612\pi\)
−0.216954 + 0.976182i \(0.569612\pi\)
\(692\) −30.9844 −1.17785
\(693\) 0 0
\(694\) 13.0800 0.496510
\(695\) 21.2410 0.805717
\(696\) 0 0
\(697\) 29.6715 1.12389
\(698\) −13.9716 −0.528831
\(699\) 0 0
\(700\) 1.70156 0.0643130
\(701\) 4.13312 0.156106 0.0780530 0.996949i \(-0.475130\pi\)
0.0780530 + 0.996949i \(0.475130\pi\)
\(702\) 0 0
\(703\) −54.1884 −2.04376
\(704\) −1.70156 −0.0641300
\(705\) 0 0
\(706\) −0.180620 −0.00679774
\(707\) −11.6326 −0.437489
\(708\) 0 0
\(709\) 38.5867 1.44915 0.724576 0.689195i \(-0.242036\pi\)
0.724576 + 0.689195i \(0.242036\pi\)
\(710\) 1.40312 0.0526583
\(711\) 0 0
\(712\) 27.0444 1.01353
\(713\) 8.25167 0.309027
\(714\) 0 0
\(715\) 4.56844 0.170850
\(716\) −8.08857 −0.302284
\(717\) 0 0
\(718\) 12.9061 0.481652
\(719\) 13.4920 0.503165 0.251583 0.967836i \(-0.419049\pi\)
0.251583 + 0.967836i \(0.419049\pi\)
\(720\) 0 0
\(721\) 9.18893 0.342214
\(722\) 24.7900 0.922588
\(723\) 0 0
\(724\) −0.590980 −0.0219636
\(725\) −0.133124 −0.00494409
\(726\) 0 0
\(727\) −15.0923 −0.559743 −0.279872 0.960037i \(-0.590292\pi\)
−0.279872 + 0.960037i \(0.590292\pi\)
\(728\) 9.23804 0.342385
\(729\) 0 0
\(730\) 6.85054 0.253550
\(731\) −25.5438 −0.944770
\(732\) 0 0
\(733\) 5.07134 0.187314 0.0936572 0.995605i \(-0.470144\pi\)
0.0936572 + 0.995605i \(0.470144\pi\)
\(734\) −12.2223 −0.451132
\(735\) 0 0
\(736\) 18.6976 0.689202
\(737\) 6.41464 0.236286
\(738\) 0 0
\(739\) −22.7021 −0.835112 −0.417556 0.908651i \(-0.637113\pi\)
−0.417556 + 0.908651i \(0.637113\pi\)
\(740\) −11.4917 −0.422443
\(741\) 0 0
\(742\) 6.46728 0.237421
\(743\) −29.2727 −1.07391 −0.536955 0.843611i \(-0.680426\pi\)
−0.536955 + 0.843611i \(0.680426\pi\)
\(744\) 0 0
\(745\) 11.5072 0.421592
\(746\) −6.55129 −0.239860
\(747\) 0 0
\(748\) 4.59688 0.168078
\(749\) −13.0199 −0.475735
\(750\) 0 0
\(751\) 7.65211 0.279229 0.139615 0.990206i \(-0.455414\pi\)
0.139615 + 0.990206i \(0.455414\pi\)
\(752\) 1.49285 0.0544387
\(753\) 0 0
\(754\) −0.332239 −0.0120994
\(755\) 7.58830 0.276167
\(756\) 0 0
\(757\) −20.9576 −0.761717 −0.380858 0.924633i \(-0.624371\pi\)
−0.380858 + 0.924633i \(0.624371\pi\)
\(758\) 9.94737 0.361305
\(759\) 0 0
\(760\) 16.2249 0.588539
\(761\) 8.16509 0.295984 0.147992 0.988989i \(-0.452719\pi\)
0.147992 + 0.988989i \(0.452719\pi\)
\(762\) 0 0
\(763\) −0.109506 −0.00396439
\(764\) 20.1429 0.728743
\(765\) 0 0
\(766\) −12.2094 −0.441143
\(767\) 4.38326 0.158270
\(768\) 0 0
\(769\) −46.3583 −1.67172 −0.835862 0.548939i \(-0.815032\pi\)
−0.835862 + 0.548939i \(0.815032\pi\)
\(770\) −0.546295 −0.0196871
\(771\) 0 0
\(772\) 7.05876 0.254050
\(773\) 24.2169 0.871021 0.435510 0.900184i \(-0.356568\pi\)
0.435510 + 0.900184i \(0.356568\pi\)
\(774\) 0 0
\(775\) 2.33897 0.0840184
\(776\) 3.61116 0.129633
\(777\) 0 0
\(778\) −13.9230 −0.499165
\(779\) 88.1241 3.15737
\(780\) 0 0
\(781\) 2.56844 0.0919060
\(782\) −5.20665 −0.186190
\(783\) 0 0
\(784\) 2.29844 0.0820871
\(785\) −12.8583 −0.458933
\(786\) 0 0
\(787\) 47.1916 1.68220 0.841100 0.540880i \(-0.181909\pi\)
0.841100 + 0.540880i \(0.181909\pi\)
\(788\) 32.9530 1.17390
\(789\) 0 0
\(790\) 1.40312 0.0499209
\(791\) −3.83844 −0.136479
\(792\) 0 0
\(793\) 59.9206 2.12784
\(794\) 5.65678 0.200752
\(795\) 0 0
\(796\) −22.5665 −0.799849
\(797\) 6.97474 0.247058 0.123529 0.992341i \(-0.460579\pi\)
0.123529 + 0.992341i \(0.460579\pi\)
\(798\) 0 0
\(799\) −1.75468 −0.0620763
\(800\) 5.29991 0.187380
\(801\) 0 0
\(802\) −0.0513358 −0.00181273
\(803\) 12.5400 0.442527
\(804\) 0 0
\(805\) −3.52790 −0.124342
\(806\) 5.83740 0.205614
\(807\) 0 0
\(808\) −23.5228 −0.827528
\(809\) 39.1428 1.37619 0.688093 0.725622i \(-0.258448\pi\)
0.688093 + 0.725622i \(0.258448\pi\)
\(810\) 0 0
\(811\) −22.1116 −0.776444 −0.388222 0.921566i \(-0.626911\pi\)
−0.388222 + 0.921566i \(0.626911\pi\)
\(812\) −0.226518 −0.00794923
\(813\) 0 0
\(814\) 3.68947 0.129316
\(815\) 22.6525 0.793481
\(816\) 0 0
\(817\) −75.8647 −2.65417
\(818\) 1.58830 0.0555337
\(819\) 0 0
\(820\) 18.6884 0.652627
\(821\) 49.1868 1.71663 0.858316 0.513122i \(-0.171511\pi\)
0.858316 + 0.513122i \(0.171511\pi\)
\(822\) 0 0
\(823\) −4.04853 −0.141123 −0.0705615 0.997507i \(-0.522479\pi\)
−0.0705615 + 0.997507i \(0.522479\pi\)
\(824\) 18.5813 0.647312
\(825\) 0 0
\(826\) −0.524151 −0.0182375
\(827\) −20.5217 −0.713610 −0.356805 0.934179i \(-0.616134\pi\)
−0.356805 + 0.934179i \(0.616134\pi\)
\(828\) 0 0
\(829\) 22.7665 0.790714 0.395357 0.918528i \(-0.370621\pi\)
0.395357 + 0.918528i \(0.370621\pi\)
\(830\) 4.23781 0.147097
\(831\) 0 0
\(832\) −7.77348 −0.269497
\(833\) −2.70156 −0.0936036
\(834\) 0 0
\(835\) 3.21795 0.111362
\(836\) 13.6527 0.472188
\(837\) 0 0
\(838\) −8.88602 −0.306963
\(839\) −9.06473 −0.312949 −0.156475 0.987682i \(-0.550013\pi\)
−0.156475 + 0.987682i \(0.550013\pi\)
\(840\) 0 0
\(841\) −28.9823 −0.999389
\(842\) −8.79790 −0.303196
\(843\) 0 0
\(844\) −4.32197 −0.148768
\(845\) 7.87063 0.270758
\(846\) 0 0
\(847\) −1.00000 −0.0343604
\(848\) −27.2099 −0.934392
\(849\) 0 0
\(850\) −1.47585 −0.0506212
\(851\) 23.8261 0.816749
\(852\) 0 0
\(853\) −13.0854 −0.448035 −0.224018 0.974585i \(-0.571917\pi\)
−0.224018 + 0.974585i \(0.571917\pi\)
\(854\) −7.16531 −0.245192
\(855\) 0 0
\(856\) −26.3280 −0.899874
\(857\) −38.4589 −1.31373 −0.656866 0.754007i \(-0.728118\pi\)
−0.656866 + 0.754007i \(0.728118\pi\)
\(858\) 0 0
\(859\) −0.435576 −0.0148617 −0.00743083 0.999972i \(-0.502365\pi\)
−0.00743083 + 0.999972i \(0.502365\pi\)
\(860\) −16.0886 −0.548616
\(861\) 0 0
\(862\) 10.5555 0.359523
\(863\) −38.7362 −1.31860 −0.659298 0.751882i \(-0.729146\pi\)
−0.659298 + 0.751882i \(0.729146\pi\)
\(864\) 0 0
\(865\) 18.2094 0.619137
\(866\) 20.2381 0.687719
\(867\) 0 0
\(868\) 3.97991 0.135087
\(869\) 2.56844 0.0871283
\(870\) 0 0
\(871\) 29.3049 0.992959
\(872\) −0.221437 −0.00749881
\(873\) 0 0
\(874\) −15.4637 −0.523068
\(875\) −1.00000 −0.0338062
\(876\) 0 0
\(877\) −34.8044 −1.17526 −0.587630 0.809130i \(-0.699939\pi\)
−0.587630 + 0.809130i \(0.699939\pi\)
\(878\) 7.90764 0.266870
\(879\) 0 0
\(880\) 2.29844 0.0774803
\(881\) −28.8583 −0.972261 −0.486130 0.873886i \(-0.661592\pi\)
−0.486130 + 0.873886i \(0.661592\pi\)
\(882\) 0 0
\(883\) −27.5988 −0.928772 −0.464386 0.885633i \(-0.653725\pi\)
−0.464386 + 0.885633i \(0.653725\pi\)
\(884\) 21.0005 0.706325
\(885\) 0 0
\(886\) 14.4302 0.484791
\(887\) −34.0939 −1.14476 −0.572380 0.819988i \(-0.693980\pi\)
−0.572380 + 0.819988i \(0.693980\pi\)
\(888\) 0 0
\(889\) 5.45518 0.182961
\(890\) −7.30621 −0.244905
\(891\) 0 0
\(892\) −34.9007 −1.16856
\(893\) −5.21140 −0.174393
\(894\) 0 0
\(895\) 4.75362 0.158896
\(896\) 11.5294 0.385169
\(897\) 0 0
\(898\) 7.51295 0.250710
\(899\) −0.311373 −0.0103849
\(900\) 0 0
\(901\) 31.9823 1.06548
\(902\) −6.00000 −0.199778
\(903\) 0 0
\(904\) −7.76188 −0.258156
\(905\) 0.347316 0.0115452
\(906\) 0 0
\(907\) −26.7979 −0.889810 −0.444905 0.895578i \(-0.646763\pi\)
−0.444905 + 0.895578i \(0.646763\pi\)
\(908\) 33.3709 1.10745
\(909\) 0 0
\(910\) −2.49571 −0.0827321
\(911\) 27.2083 0.901451 0.450726 0.892663i \(-0.351165\pi\)
0.450726 + 0.892663i \(0.351165\pi\)
\(912\) 0 0
\(913\) 7.75737 0.256731
\(914\) −5.45460 −0.180422
\(915\) 0 0
\(916\) −34.7510 −1.14820
\(917\) −21.1084 −0.697062
\(918\) 0 0
\(919\) 1.61134 0.0531533 0.0265767 0.999647i \(-0.491539\pi\)
0.0265767 + 0.999647i \(0.491539\pi\)
\(920\) −7.13393 −0.235199
\(921\) 0 0
\(922\) −1.81321 −0.0597149
\(923\) 11.7338 0.386221
\(924\) 0 0
\(925\) 6.75362 0.222058
\(926\) −11.0202 −0.362146
\(927\) 0 0
\(928\) −0.705544 −0.0231606
\(929\) −53.7396 −1.76314 −0.881570 0.472053i \(-0.843513\pi\)
−0.881570 + 0.472053i \(0.843513\pi\)
\(930\) 0 0
\(931\) −8.02362 −0.262963
\(932\) −29.4888 −0.965936
\(933\) 0 0
\(934\) −2.01715 −0.0660030
\(935\) −2.70156 −0.0883505
\(936\) 0 0
\(937\) −26.0601 −0.851348 −0.425674 0.904877i \(-0.639963\pi\)
−0.425674 + 0.904877i \(0.639963\pi\)
\(938\) −3.50429 −0.114419
\(939\) 0 0
\(940\) −1.10518 −0.0360469
\(941\) 57.6176 1.87828 0.939139 0.343537i \(-0.111625\pi\)
0.939139 + 0.343537i \(0.111625\pi\)
\(942\) 0 0
\(943\) −38.7473 −1.26178
\(944\) 2.20527 0.0717755
\(945\) 0 0
\(946\) 5.16531 0.167939
\(947\) −58.3245 −1.89529 −0.947646 0.319323i \(-0.896545\pi\)
−0.947646 + 0.319323i \(0.896545\pi\)
\(948\) 0 0
\(949\) 57.2882 1.85965
\(950\) −4.38326 −0.142212
\(951\) 0 0
\(952\) −5.46295 −0.177055
\(953\) −2.27670 −0.0737496 −0.0368748 0.999320i \(-0.511740\pi\)
−0.0368748 + 0.999320i \(0.511740\pi\)
\(954\) 0 0
\(955\) −11.8379 −0.383064
\(956\) −19.8068 −0.640597
\(957\) 0 0
\(958\) −22.4263 −0.724559
\(959\) −3.40312 −0.109893
\(960\) 0 0
\(961\) −25.5292 −0.823523
\(962\) 16.8551 0.543430
\(963\) 0 0
\(964\) 38.3582 1.23544
\(965\) −4.14840 −0.133542
\(966\) 0 0
\(967\) 51.7348 1.66368 0.831840 0.555016i \(-0.187288\pi\)
0.831840 + 0.555016i \(0.187288\pi\)
\(968\) −2.02214 −0.0649942
\(969\) 0 0
\(970\) −0.975577 −0.0313239
\(971\) 21.6846 0.695893 0.347947 0.937514i \(-0.386879\pi\)
0.347947 + 0.937514i \(0.386879\pi\)
\(972\) 0 0
\(973\) −21.2410 −0.680955
\(974\) −4.41626 −0.141506
\(975\) 0 0
\(976\) 30.1468 0.964975
\(977\) 15.7235 0.503041 0.251520 0.967852i \(-0.419069\pi\)
0.251520 + 0.967852i \(0.419069\pi\)
\(978\) 0 0
\(979\) −13.3741 −0.427438
\(980\) −1.70156 −0.0543544
\(981\) 0 0
\(982\) 1.89754 0.0605530
\(983\) −26.0134 −0.829699 −0.414849 0.909890i \(-0.636166\pi\)
−0.414849 + 0.909890i \(0.636166\pi\)
\(984\) 0 0
\(985\) −19.3663 −0.617063
\(986\) 0.196471 0.00625690
\(987\) 0 0
\(988\) 62.3714 1.98430
\(989\) 33.3570 1.06069
\(990\) 0 0
\(991\) 39.8932 1.26725 0.633624 0.773641i \(-0.281567\pi\)
0.633624 + 0.773641i \(0.281567\pi\)
\(992\) 12.3963 0.393584
\(993\) 0 0
\(994\) −1.40312 −0.0445044
\(995\) 13.2622 0.420441
\(996\) 0 0
\(997\) 10.9914 0.348102 0.174051 0.984737i \(-0.444314\pi\)
0.174051 + 0.984737i \(0.444314\pi\)
\(998\) −15.8818 −0.502731
\(999\) 0 0
Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))

Twists

       By twisting character
Char Parity Ord Type Twist Min Dim
1.1 even 1 trivial 3465.2.a.bl.1.3 4
3.2 odd 2 1155.2.a.u.1.2 4
15.14 odd 2 5775.2.a.bz.1.3 4
21.20 even 2 8085.2.a.bn.1.2 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
1155.2.a.u.1.2 4 3.2 odd 2
3465.2.a.bl.1.3 4 1.1 even 1 trivial
5775.2.a.bz.1.3 4 15.14 odd 2
8085.2.a.bn.1.2 4 21.20 even 2